21.0 Two-Factor Designs

• Answer Questions

• RCBD

• Concrete Example

• Two-Way ANOVA

• Popcorn Example

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21.4 RCBD

The Randomized Complete Block Design is also known as the two-way

ANOVA without interaction. A key assumption in the analysis is that the

effect of each level of the treatment factor is the same for each level of the

blocking factor.

That assumption would be violated if, say, a particular fertilizer worked well

for one stain but poorly for another; or if one cancer therapy were better for

lung cancer but a different therapy were better for stomach cancer.

In RCBD, there is one observation for each combination of levels of the

treatment and block factors.

2

RCBD notation:

• I is the number of treatments; J is the number of blocks

• Xij is the measurement on the unit in block j that received treatment i.

• n.. = I ∗ J is the total number of experimental units.

• ni. = J and n.j = I, the number of observations for a given treatment

level or block level, respectively.

• Xi. is the sum of all measurements for units receiving treatment i, and X.j

is the sum of all measurements for units in the j block.

• X̄i. is the average of all measurements for units receiving treatment i, or1

J

∑Jj=1

Xij .

• X̄.j is the average of all measurements for units in the jth block, or1

I

∑Ii=1

Xij .

• X̄.. is the average of all measurements.

We continue to use the dot and bar conventions.

3

Some assumptions:

The model for an RCBD (or two-way ANOVA without interactions) is:

Xij = µ+ τi + βj + ǫij

where µ is the overall mean of all experimental units, τi is the effect of

treatment i, βj is the effect of block j, and the ǫij are random errors that are:

• normally distributed with mean zero and unknown standard deviation σ

• independent of the values for all other errors.

Note how this generalizes the one-way ANOVA model:

Xij = µ+ τi + ǫij.

4

There are two hypothesis tests in an RCBD, and they are always the same:

H0: The means of all treatments are equal or H0: τ1 = · · · = τI = 0

versus

HA: At least one of the treatments has a different mean

and

H0: The means of all blocks are equal or H0: β1 = · · · = βJ = 0.

versus

HA: At least one of the blocks has a different mean

In the case of an RCBD, we divide the total sum of squares (SStot) into the

part attributable to differences between the treatment means (SStrt), the part

attributable to difference between the blocks (SSblk) and the part attributable

to differences within block-treatment groups, or the sum of squares due to

pure error (SSerr).

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The test statistics for an RCBD look at the standardized ratio of the

between-group sum of squares to the within-group sum of squares for the

treatment and block effects separately. If one or both are large, then it suggest

that some of the treatment or block effects are not equal.

Formally, the test statistics are:

tstrt =SStrt/(I − 1)

SSerr/(IJ − I − J + 1)

and

tsblk =SSblk/(J − 1)

SSerr/(IJ − I − J + 1).

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These two test statistics are compared to an F distribution with I − 1 or J − 1

(respectively) degrees of freedom in the numerator and IJ − I − J + 1 degrees

of freedom in the denominator.

The numerator degrees of freedom for the test of treatments is I − 1. That

is because we make I − 1 estimates to get it; one estimate for each of the I

different groups, but the last one is not needed since the sum of the treatment

effects is forced to add up to zero (since we have already found the overall

mean, which cost us one 1 degree of freedom).

Similarly, the numerator degrees of freedom for the block term is J − 1.

Since that the total degrees of freedom is IJ − 1, then the

degrees of freedom for the error term in the denominator must be

(IJ − 1)− (I − 1)− (J − 1) = IJ − I − J + 1 = (I − 1)(J − 1). As always, we

have lost one degree of freedom due to estimating µ, the overall mean of the

combined populations.

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To understand this test statistic, consider the following definitional formulae:

SStot =I∑

i=1

J∑

j=1

(Xij − X̄..)2

SStrt =I∑

i=1

J(X̄i. − X̄..)2

SSblk =J∑

j=1

I(X̄.j − X̄..)2

SSerr =

I∑

i=1

J∑

j=1

(Xij − X̄i. − X̄.j + X̄..)2

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In practice, if we aren’t using a computer, then it is easier, faster, and more

numerically stable to use the following computational formulae:

SStot =

I∑

i=1

J∑

j=1

X2

ij

−

X2..

IJ

SSblk =J∑

j=1

X2

.j

I−

X2..

IJ

SStrt =

I∑

i=1

X2

i.

J−

X2..

IJ

SSerr =

I∑

i=1

J∑

j=1

X2

ij

−

I∑

i=1

X2

i.

J−

J∑

j=1

X2

.j

I+

X2..

IJ

9

Note that SStot = SStrt + SSblk + SSerr. This is useful because it simplifies

calculation—if you know three of the sum of square terms, you can find the

fourth by subtraction.

Once you have calculated the test statistics, you refer them to F distributions.

For the test of equal treatment effects, the numerator df is I − 1. For the test

of equal block effects, the numerator df is J − 1. Both tests have the same

denominator df, (I − 1)(J − 1).

Note that both RCBD tests are one-sided—we reject if and only if we get large

values of the test statistic. Also note that if we had only two treatments, then

this would be informationally equivalent to a paired difference two-sample

t-test and use an F with 1 df in the numerator and J−1 df in the denominator.

10

To simplify the organization of the calculations in an RCBD, it is customary

to write things in a table.

Source df SS MS F

treatment I − 1 SStrt SStrt/(I − 1) MStrt/MSerr

block J − 1 SSblk SSblk/(J − 1) MSblk/MSerr

error (I − 1)(J − 1) SSerr SSerr/(I − 1)(J − 1)

total IJ − 1 SStot

The MS column contains the Mean Squares, which are the average sum

of squares attributable to each component in the partition. The F column

contains the test statistic.

What happens if the block effect is not significant?

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21.2 Concrete Example

Suppose you are manufacturing concrete cylinders for, say, bridge supports.

There are three ways of drying green concrete (say A, B, and C), and you want

to find the one that gives you the best compressive strength. The concrete is

mixed in batches that are large enough to produce exactly three cylinders, and

your production engineer believes that there is substantial variation in the

quality of the concrete from batch to batch.

You have data from J = 5 batches on each of the I = 3 drying processes. Your

measurements are the compressive strength of the cylinder in a destructive

test. (So there is an economic incentive to learn as much as you can from a

well-designed experiment.)

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The data are:

Batch

Treatment 1 2 3 4 5 Trt Sum

A 52 47 44 51 42 236

B 60 55 49 52 43 259

C 56 48 45 44 38 231

Batch Mean 168 150 138 147 123 726

The primary null hypothesis is that all three drying techniques are equivalent,

in terms of conferring compressive strength. The secondary null is that

the batches are equivalent (but if they are, then we have wasted power by

controlling for an effect that is small or non-existent).

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We use the computational forms for the sum of squares calculation. Thus:

SStot =

I∑

i=1

J∑

j=1

X2

ij

−

X2..

IJ

= 499.6

SSblk =

J∑

j=1

X2

.j

I−

X2..

IJ

= 363.6

SStrt =I∑

i=1

X2

i.

J−

X2..

IJ

= 89.2

SSerr =

I∑

i=1

J∑

j=1

X2

ij

−

I∑

i=1

X2

i.

J−

J∑

j=1

X2

.j

I+

X2..

IJ

= 46.8

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We plug this into the ANOVA table:

Source df SS MS F

drying 2 89.2 44.6 7.62

batch 4 363.6 90.9 15.54

error 8 46.8 5.85

total 14 499.6

Our test statistics are 7.62 and 15.54. The test of the first uses an F with 2 df

in the numerator, 8 in the denominator, so the 0.05 critical value is 4.46. We

reject.

The secondary test has 4 df in the numerator, 8 in the denominator, and the

0.05 critical value is 3.84. The blocking was a good idea.

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Suppose we had not blocked for batch. Then the data would be:

Treatment Trt Sum

A 52, 47, 44, 51, 42 236

B 60, 55, 49, 52, 43 259

C 56, 48, 45, 44, 38 231

This is the same as before except now we ignore which batch the observation

came from.

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The one-way ANOVA table for this is:

Source df SS MS F

drying 2 89.2 44.6 1.30

error 4 + 8 46.8+363.6 34.2

total 14 499.6

Note that this gives a test statistic of 1.30, which is referred to an

F-distribution with 2 df in the numerator, 12 in the denominator. The .05

critical value is 3.89. We fail to reject the null.

Using blocks gave us a more powerful test.

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21.3 Two-Way ANOVA

The two-way Analysis of Variance is used when one suspects that there may

be some kind of interaction between the levels of two experimental factors.

This would occur, for example, if

• one kind of fertilizer were best for sunny fields, and a different kind for

wet fields;

• one kind of chemotherapy was better for lung cancer, and a different kind

for stomach cancer;

• female students learned better from reading assignments, but male

students learned better from lecture.

The presence of interaction means that the simple RCBD model, with a block

effect and a treatment effect that add, cannot explain the observations.

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Two-Way ANOVA notation:

• I (J) is the number of levels of Factor A (B).

• Xijk is the measurement on the kth experimental unit that receives level i

of factor A and level j of factor B.

• the index k runs from 1 to nij., the number of units that receive level i of

factor and level j of factor B.

• n... is the total number of experimental units.

• ni.. and n.j. are the number of units receiving the ith or j levels of factors

A or B, respectively. And nij. is the number of units that receive level i of

factor A and level j of factor B.

• Xi... is the sum of all measurements for units receiving level i of factor A;

X.j. is the sum of all units receiving level j of factor B; Xij. is the sum for

all units receiving both.

We continue to use the dot and bar conventions. So X̄i.. is the average of all

measurements on units receiving level i of factor A, and so forth.

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The model for the two-way ANOVA with interaction is:

Xijk = µ+ αi + βj + γij + ǫijk.

where γij is the interaction between levels i and j of factors A and B, and the

errors are:

• normally distributed with mean zero and unknown standard deviation σ

• independent of the values for all other errors.

As written, this model is not identifiable. To fit it, we add the additional

constraints that:

α1 + · · ·+ αI = 0

β1 + · · ·+ βJ = 0

γ11 + · · ·+ γIJ = 0

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There are three hypothesis tests in a two-way ANOVA with interaction, and

they are always the same:

H0: The interaction effects are all zero or H0: γ11 = · · · = γIJ = 0

versus

HA: At least one of the interactions is non-zero.

The next two tests concern factors A and B:

H0: The means of all levels of Factor A are equal or

H0: α1 = · · · = αI = 0.

versus

HA: At least one of the factor A levels has a non-zero mean

and similarly for the corresponding hypotheses for Factor B.

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If the interaction term is statistically significant, we usually immediately

decide that both main effects (Factors A and B) are also significant, no matter

what the result of the test.

It can happen that both main effects tests are not significant but the

interaction is. This situation is called masking.

As usual, we divide the total sum of squares (SStot) into the parts attributable

to differences between the treatment means or main effects, written as SSAand SSB, the part attributable to the interaction, written as SSAB, and the

part attributable to random variation, or SSerr.

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For the fixed effects model, the test statistics are:

tsA =SSA/(I − 1)

SSerr/(n... − IJ)

tsB =SSB/(J − 1)

SSerr/(n... − IJ)

tsAB =SSAB/(I − 1)(J − 1)

SSerr/(n... − IJ)

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The test statistics for main effects A and B, and interaction AB, are compared

to an F distribution with I − 1 or J − 1 or (I − 1)(J − 1) degrees of freedom,

respectively, in the numerator and n... − IJ degrees of freedom in the

denominator.

As usual, thoughtful counting will show that the degrees of freedom correspond

to the number of estimates that one makes to find the effects of each of these

quantities.

Since that the total degrees of freedom is n... − 1, then the

degrees of freedom for the error term in the denominator must be

n... − (I − 1)− (J − 1)− (I − 1)(J − 1)− 1 = n... − IJ . As always, we have lost

one degree of freedom due to estimating µ, the overall mean of the combined

populations.

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The following definitional formulae for the sums of squares terms are:

SStot =

I∑

i=1

J∑

j=1

nij.∑

k=1

(Xijk − X̄...)2

SSA =I∑

i=1

J∑

j=1

nij.∑

k=1

(X̄i.. − X̄...)2

SSB =I∑

i=1

J∑

j=1

nij.∑

k=1

(X̄.j. − X̄...)2

SSAB =

I∑

i=1

J∑

j=1

nij∑

k=1

(X̄ij. − X̄i.. − X̄.j. + X̄...)2

One finds SSerr by subtraction.

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To simplify the organization of the calculations in a fixed effects two-way

ANOVA with interaction, it is customary to write things in a table.

Source df SS MS F

treatment A I − 1 SSA SSA/(I − 1) MSA/MSerr

treatment B J − 1 SSB SSB/(J − 1) MSB/MSerr

interaction (I − 1)(J − 1) SSAB SSAB/(I − 1)(J − 1) MSAB/MSerr

error n... − IJ SSerr SSerr/(n... − IJ)

total n... − 1 SStot

The MS column contains the Mean Squares, which are the average sum

of squares attributable to each component in the partition. The F column

contains the test statistic.

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21.4 Popcorn Example

Suppose you want to compare type of popcorn popper and brand of popcorn

with respect to their yield (in terms of cups of popped corn). Factor A is the

type of popper: oil-based versus air-based. Factor B is the brand of popcorn:

gourmet versus national brand versus generic. For each combination of popper

type and brand, you took three separate measurements.

Corn

Popper Gourmet Nat’l Brand Generic Row Sum

Oil 5.5, 5.5, 6 4.5, 4.5, 4 3.5, 4, 3 40.5

Air 6.5, 7, 7 5, 5.5, 5 4, 5, 4.5 49.5

col sum 37.5 28.5 24.0 90.0

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We use the computational forms for the sum of squares calculation with n the

number of observations for each combination of factor levels.

SStot =

I∑

i=1

J∑

j=1

nij.∑

k=1

X2

ijk

−

X2...

n...

= 22.0

SSA =1

Jn

(

I∑

i=1

X2

i..

)

−

X2...

n...

= 4.5

SSB =1

In

J∑

j=1

X2

.j.

−

X2...

n...

= 15.75

SSAB =1

n

I∑

i=1

J∑

j=1

X2

ij.

−

X2...

n...

− SSA − SSB = .083

SSerr = SStot − SSA − SSB − SSAB = 1.667

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We plug this into the ANOVA table:

Source df SS MS F

popper 1 4.5 4.5 32.4

corn 2 15.75 7.87 56.7

interaction 2 .083 .042 .3

error 12 1.667 .139

total 17 22.0

Our test statistics are .3 for the interaction, 56.7 for the corn, and 32.4 for the

popper. The interaction test and the corn test both use an F with 2 df in

the numerator, 12 in the denominator, so the 0.05 critical value is 3.89. We

conclude that there is no interaction and that the brand of corn is significantly

different. For the popper, there is 1 df in the numerator and the F critical

value is 4.75. There is strong evidence that the popper is also significantly

different.

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